It is equivalent to consider sampling from \(\mathcal {N}(0,{\mathbf {I}})\)  and then divide it by its norm.
By the initialization condition, we apply the Gaussian concentration bound and Theorem 3.1.1 in [1]} and get
\(\Pr [({\mathbf {x}}_0^\top {\mathbf {y}}_0)^2<\frac{\delta _{0}^2}{3d}]< \Pr [{\mathbf {x}}_1^2<\delta _0^2]+\Pr [\Vert {\mathbf {x}}\Vert ^2<1.5d] < 2\delta _0+2\exp \lbrace -\Omega (d)\rbrace \) 
